Stop character from falling/sticking to walls

Hello,

I am trying to make my character hang on a wall but I am facing 2 problems.

First problem is how do I know where my character is hanging at? The bottom, left/right side. I am not sure how to get started with this.

My first thought was to put 3 different colliders and give them a name depending on the side there are on. But not sure if this is a good idea performance wise.

Second problem I encounter is how do I stop the character from falling down? I figured this would be the easy part but I am stuck with it as I can't figure out how to make a rigid body stop falling.

I have no idea how to handle the not falling part. First I tried to set gravity to zero on the object but that doesn't work.

The code so far I have although very trail and error:

 public class MovementScript : MonoBehaviour
 {
 
     public float maxSpeed = 5f;
 
     public bool startJump = false;
     public bool isJumping = false;
     public bool isFalling = false;
 
     public float jumpForce = 150f;
 
     //public LayerMask whatIsGround;
 
     public bool hangingOnWall = false;
     public bool jumpOffWall = false;
     public float jumpDirection;
 
     void Start ()
     {
     
     }
 
     void FixedUpdate()
     {
         float move = Input.GetAxis("Horizontal");
 
         rigidbody2D.velocity = new Vector2 (move * maxSpeed, rigidbody2D.velocity.y);
 
         if(Input.GetButton ("Jump") && (!isFalling && !isJumping ) )
         {
             startJump = true;
         }
 
         if(hangingOnWall && move != 0)
         {
             jumpOffWall = true;
             jumpDirection = move;
         }
 
     }
 
     void Update ()
     {
         if(startJump)
         {
             rigidbody2D.AddForce(new Vector2(0,jumpForce));
             startJump = false;
         }
 
         if (rigidbody2D.velocity.y > 0)
         {
             isJumping = true;
             isFalling = false;
         }
         else if(rigidbody2D.velocity.y < 0)
         { 
             isJumping = false;
             isFalling = true;
         }
         else
         {
             isJumping = false;
             isFalling = false;
         }
 
         if(hangingOnWall)
         {
         }
 
         if(jumpOffWall)
         {
             rigidbody2D.AddForce(new Vector2(jumpDirection,jumpForce));
             jumpOffWall = false;
             hangingOnWall = false;
             jumpDirection = 0f;
         }
     }
 
     void OnCollisionEnter2D(Collision2D collision)
     {
         if (collision.collider.gameObject.layer == LayerMask.NameToLayer("Ground"))
         {
             Debug.Log("on the ground");
         }
         if (collision.collider.gameObject.layer == LayerMask.NameToLayer("Walls"))
         {
             if(rigidbody2D.velocity.x == 0)
             {
                 Debug.Log (rigidbody2D.velocity);
                 hangingOnWall = true;
             }
         }
     }
 }

Thanks in advance!